MCMC methods [1]} are based on constructing a Markov chain that has the target distribution as an equilibrium distribution. Independent Metropolis Hastings uses a state independent - usually fixed - proposal for each step. In our case, it only requires applying a transformation and computing a forward pass of the neural network to evaluate the loss. This enables the use of non-differentiable transformations, and has the advantage that the density at consecutive proposals can be evaluated in parallel allowing speedups in the sampling step. Although MH methods thus allow to sample the proposal distribution with low computational cost, they exhibit random walk behaviour, which leads to slow convergence in high dimensional settings [2]}, [3]}. However, a uniform augmentation distribution has been shown to give good results [4]} and sampling \(\lambda ^{*}_c\)  accurately is not a concern given that we are interested in promoting approximate invariance.
